On the Iwasawa main conjecture for the double product.

Let \sigma and \tau denote a pair of absolutely irreducible p-ordinary and p-distinguished Galois representations into \operatorname {GL}_2(\overline {\mathbb {F}}_p). Given two primitive forms (f,g) such that \operatorname {wt}(f)>\operatorname {wt}(g)> 1 and where \overline {\rho }_f\cong \sigma and \overline {\rho }_g\cong \tau, we show that the Iwasawa Main Conjecture for the double product \rho _f\otimes \rho _g depends only on the residual Galois representation \sigma \otimes \tau : G_{\mathbb {Q}}\rightarrow \operatorname {GL}_4(\overline {\mathbb {F}}_p). More precisely, if IMC(f\otimes g) is true for one pair (f,g) with \overline {\rho }_f \cong \sigma and \overline {\rho }_g\cong \tau and whose \mu-invariant equals zero, then it is true for every congruent pair too. [ABSTRACT FROM AUTHOR]